《视觉 SLAM 十四讲》V2 第 9 讲后端优化1 【扩展卡尔曼滤波器 EKF BA+非线性优化(Ceres、g2o)】

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《视觉 SLAM 十四讲》V2 第 9 讲后端优化1 【扩展卡尔曼<a href=https://www.elefans.com/category/jswz/34/1762689.html style= 滤波器 EKF BA+非线性优化(Ceres、g2o)】"/>

《视觉 SLAM 十四讲》V2 第 9 讲后端优化1 【扩展卡尔曼滤波器 EKF BA+非线性优化(Ceres、g2o)】

文章目录

第9讲后端1
- - 9.1.2 线性系统和 KF
  - 9.1.4 扩展卡尔曼滤波器 EKF 不足
- 9.2 BA 与图优化
- - 9.2.1 投影模型和 BA 代价函数
  - 9.2.2 BA 的求解
  - 9.2.3 稀疏性和边缘化
  - 9.2.4 鲁棒核函数
- 9.3 实践： Ceres BA 【Code】
- - - - 本讲 CMakeLists.txt
- 9.4 实践：g2o 求解 BA 【Code】
- 习题
- - √ 题 1

第9讲后端1

滤波器 EKF

前端视觉里程计：短时间内的轨迹和地图。
后端优化：长时间内的最优轨迹和地图

9.1 概述
9.1.1 状态估计的概率解释

渐进的(Incremental)：当前的状态只由过去的时刻决定，甚至只由前一个决定。
批量的(Batch)：不仅使用过去的信息更新自己的状态，也会用未来的信息来更新。

SfM: Structure from Motion.

马尔可夫性： k k k 时刻状态仅与 k − 1 k-1 k−1 时刻状态有关。【扩展卡尔曼滤波(EKF)】
考虑 k k k 时刻状态与之前所有状态的关系。【非线性优化】

视觉 SLAM 主流

9.1.2 线性系统和 KF

经典卡尔曼滤波：从概率角度出发的最大后验概率估计方式。

在线性高斯系统中，卡尔曼滤波器构成了该系统中的最大后验概率估计。
卡尔曼滤波器构成了线性系统的最优无偏估计。

9.1.3 非线性系统和 EKF

把卡尔曼滤波器的结果扩展到非线性系统中， 扩展卡尔曼滤波器。

9.1.4 扩展卡尔曼滤波器 EKF 不足

同等计算量的情况下，非线性优化能取得更好的效果。
精度和鲁棒性更好。

9.2 BA 与图优化

视觉三维重建

Bundle Adjustment (BA)：从视觉图像中提炼出最优的3D模型和相机参数(内参和外差)。
考虑从任意特征点 发射出来的几束光线(bundles of light rays)，它们会在几个相机的成像平面上变成像素或是检测到的特征点，如果我们调整(adjustment) 各相机姿态和各自特征点的空间位置，使得这些光纤最终收束到相机的光心，称为 BA

其它译法：光束法平差、捆集调整

9.2.1 投影模型和 BA 代价函数

9.2.2 BA 的求解

9.2.3 稀疏性和边缘化

Schur消元 Marginalization(边缘化)

9.2.4 鲁棒核函数

1、构造 BA 问题
2、设置 Schur 消元
3、调用稠密或稀疏矩阵求解器对变量进行优化

9.3 实践： Ceres BA 【Code】

定义投影误差模型

sudo apt-get install meshlab

本讲 CMakeLists.txt

cmake_minimum_required(VERSION 2.8)set(CMAKE_CXX_STANDARD 17)
project(bundle_adjustment)
set(CMAKE_BUILD_TYPE "Release")
set(CMAKE_CXX_FLAGS "-O3 -std=c++14")LIST(APPEND CMAKE_MODULE_PATH ${PROJECT_SOURCE_DIR}/cmake)Find_Package(G2O REQUIRED)
Find_Package(Eigen3 REQUIRED)
Find_Package(Ceres REQUIRED)
Find_Package(Sophus REQUIRED)
Find_Package(CSparse REQUIRED)SET(G2O_LIBS g2o_csparse_extension g2o_stuff g2o_core cxsparse)include_directories(${PROJECT_SOURCE_DIR} ${EIGEN3_INCLUDE_DIR} ${CSPARSE_INCLUDE_DIR})add_library(bal_common common.cpp)add_executable(bundle_adjustment_ceres bundle_adjustment_ceres.cpp)
target_link_libraries(bundle_adjustment_ceres ${CERES_LIBRARIES} bal_common)add_executable(bundle_adjustment_g2o bundle_adjustment_g2o.cpp)
target_link_libraries(bundle_adjustment_g2o ${G2O_LIBS} g2o_solver_csparse g2o_csparse_extension${Sophus_LIBRARIES}bal_common)

mkdir build && cd build
cmake ..
make 
./bundle_adjustment_ceres ../problem-16-22106-pre.txt

byzanz-record -x 72 -y 64 -w 1848 -h 893  -d 20 --delay=5 -c  /home/xixi/myGIF/test.gif

byzanz-record -x 72 -y 64 -w 1848 -h 893  -d 30 --delay=5 -c  /home/xixi/myGIF/test.gif

bundle_adjustment_ceres.cpp

#include <iostream>
#include <ceres/ceres.h>
#include "common.h"
#include "SnavelyReprojectionError.h"using namespace std;void SolveBA(BALProblem &bal_problem);int main(int argc, char **argv) {if (argc != 2) {cout << "usage: bundle_adjustment_ceres bal_data.txt" << endl;return 1;}BALProblem bal_problem(argv[1]);bal_problem.Normalize();bal_problem.Perturb(0.1, 0.5, 0.5);bal_problem.WriteToPLYFile("initial.ply");SolveBA(bal_problem);bal_problem.WriteToPLYFile("final.ply");return 0;
}void SolveBA(BALProblem &bal_problem) {const int point_block_size = bal_problem.point_block_size();const int camera_block_size = bal_problem.camera_block_size();double *points = bal_problem.mutable_points();double *cameras = bal_problem.mutable_cameras();// Observations is 2 * num_observations long array observations// [u_1, u_2, ... u_n], where each u_i is two dimensional, the x// and y position of the observation.const double *observations = bal_problem.observations();ceres::Problem problem;for (int i = 0; i < bal_problem.num_observations(); ++i) {ceres::CostFunction *cost_function;// Each Residual block takes a point and a camera as input// and outputs a 2 dimensional Residualcost_function = SnavelyReprojectionError::Create(observations[2 * i + 0], observations[2 * i + 1]);// If enabled use Huber's loss function.ceres::LossFunction *loss_function = new ceres::HuberLoss(1.0);// Each observation corresponds to a pair of a camera and a point// which are identified by camera_index()[i] and point_index()[i]// respectively.double *camera = cameras + camera_block_size * bal_problem.camera_index()[i];double *point = points + point_block_size * bal_problem.point_index()[i];problem.AddResidualBlock(cost_function, loss_function, camera, point);}// show some information here ...std::cout << "bal problem file loaded..." << std::endl;std::cout << "bal problem have " << bal_problem.num_cameras() << " cameras and "<< bal_problem.num_points() << " points. " << std::endl;std::cout << "Forming " << bal_problem.num_observations() << " observations. " << std::endl;std::cout << "Solving ceres BA ... " << endl;ceres::Solver::Options options;options.linear_solver_type = ceres::LinearSolverType::SPARSE_SCHUR;options.minimizer_progress_to_stdout = true;ceres::Solver::Summary summary;ceres::Solve(options, &problem, &summary);std::cout << summary.FullReport() << "\n";
}

9.4 实践：g2o 求解 BA 【Code】

报错：

/home/xixi/Downloads/slambook2-master/ch9/bundle_adjustment_g2o.cpp:10:10: fatal error: sophus/se3.hpp: No such file or directory10 | #include "sophus/se3.hpp"|          ^~~~~~~~~~~~~~~~

SO3d 去掉 d 。共3个
报错2：

/usr/local/include/g2o/stuff/tuple_tools.h:41:46: error: ‘tuple_size_v’ is not a member of ‘std’; did you mean ‘tuple_size’?41 |       f, t, i, std::make_index_sequence<std::tuple_size_v<std::decay_t<T>>>());

改动3：

第143-147行更换成下述代码

    std::unique_ptr<g2o::BlockSolverX::LinearSolverType> linearSolver (new g2o::LinearSolverCSparse<g2o::BlockSolverX::PoseMatrixType>());std::unique_ptr<g2o::BlockSolverX> solver_ptr (new g2o::BlockSolverX(std::move(linearSolver)));g2o::OptimizationAlgorithmLevenberg* solver = new g2o::OptimizationAlgorithmLevenberg(std::move(solver_ptr));

其它问题：链接库的问题，看 CMakeLists.txt

cd build
cmake ..
make 
./bundle_adjustment_g2o ../problem-16-22106-pre.txt

byzanz-record -x 72 -y 64 -w 1848 -h 893  -d 30 --delay=5 -c  /home/xixi/myGIF/test.gif

byzanz-record -x 72 -y 64 -w 1848 -h 893  -d 20 --delay=5 -c  /home/xixi/myGIF/test.gif

bundle_adjustment_g2o.cpp

#include <g2o/core/base_vertex.h>
#include <g2o/core/base_binary_edge.h>
#include <g2o/core/block_solver.h>
#include <g2o/core/optimization_algorithm_levenberg.h>
#include <g2o/solvers/csparse/linear_solver_csparse.h>
#include <g2o/core/robust_kernel_impl.h>
#include <iostream>#include "common.h"
#include "sophus/se3.h"using namespace Sophus;
using namespace Eigen;
using namespace std;/// 姿态和内参的结构
struct PoseAndIntrinsics {PoseAndIntrinsics() {}/// set from given data addressexplicit PoseAndIntrinsics(double *data_addr) {rotation = SO3::exp(Vector3d(data_addr[0], data_addr[1], data_addr[2]));translation = Vector3d(data_addr[3], data_addr[4], data_addr[5]);focal = data_addr[6];k1 = data_addr[7];k2 = data_addr[8];}/// 将估计值放入内存void set_to(double *data_addr) {auto r = rotation.log();for (int i = 0; i < 3; ++i) data_addr[i] = r[i];for (int i = 0; i < 3; ++i) data_addr[i + 3] = translation[i];data_addr[6] = focal;data_addr[7] = k1;data_addr[8] = k2;}SO3 rotation;Vector3d translation = Vector3d::Zero();double focal = 0;double k1 = 0, k2 = 0;
};/// 位姿加相机内参的顶点，9维，前三维为so3，接下去为t, f, k1, k2
class VertexPoseAndIntrinsics : public g2o::BaseVertex<9, PoseAndIntrinsics> {
public:EIGEN_MAKE_ALIGNED_OPERATOR_NEW;VertexPoseAndIntrinsics() {}virtual void setToOriginImpl() override {_estimate = PoseAndIntrinsics();}virtual void oplusImpl(const double *update) override {_estimate.rotation = SO3::exp(Vector3d(update[0], update[1], update[2])) * _estimate.rotation;_estimate.translation += Vector3d(update[3], update[4], update[5]);_estimate.focal += update[6];_estimate.k1 += update[7];_estimate.k2 += update[8];}/// 根据估计值投影一个点Vector2d project(const Vector3d &point) {Vector3d pc = _estimate.rotation * point + _estimate.translation;pc = -pc / pc[2];double r2 = pc.squaredNorm();double distortion = 1.0 + r2 * (_estimate.k1 + _estimate.k2 * r2);return Vector2d(_estimate.focal * distortion * pc[0],_estimate.focal * distortion * pc[1]);}virtual bool read(istream &in) {}virtual bool write(ostream &out) const {}
};class VertexPoint : public g2o::BaseVertex<3, Vector3d> {
public:EIGEN_MAKE_ALIGNED_OPERATOR_NEW;VertexPoint() {}virtual void setToOriginImpl() override {_estimate = Vector3d(0, 0, 0);}virtual void oplusImpl(const double *update) override {_estimate += Vector3d(update[0], update[1], update[2]);}virtual bool read(istream &in) {}virtual bool write(ostream &out) const {}
};class EdgeProjection :public g2o::BaseBinaryEdge<2, Vector2d, VertexPoseAndIntrinsics, VertexPoint> {
public:EIGEN_MAKE_ALIGNED_OPERATOR_NEW;virtual void computeError() override {auto v0 = (VertexPoseAndIntrinsics *) _vertices[0];auto v1 = (VertexPoint *) _vertices[1];auto proj = v0->project(v1->estimate());_error = proj - _measurement;}// use numeric derivativesvirtual bool read(istream &in) {}virtual bool write(ostream &out) const {}};void SolveBA(BALProblem &bal_problem);int main(int argc, char **argv) {if (argc != 2) {cout << "usage: bundle_adjustment_g2o bal_data.txt" << endl;return 1;}BALProblem bal_problem(argv[1]);bal_problem.Normalize();bal_problem.Perturb(0.1, 0.5, 0.5);bal_problem.WriteToPLYFile("initial.ply");SolveBA(bal_problem);bal_problem.WriteToPLYFile("final.ply");return 0;
}void SolveBA(BALProblem &bal_problem) {const int point_block_size = bal_problem.point_block_size();const int camera_block_size = bal_problem.camera_block_size();double *points = bal_problem.mutable_points();double *cameras = bal_problem.mutable_cameras();// pose dimension 9, landmark is 3/*typedef g2o::BlockSolver<g2o::BlockSolverTraits<9, 3>> BlockSolverType;typedef g2o::LinearSolverCSparse<BlockSolverType::PoseMatrixType> LinearSolverType;// use LMauto solver = new g2o::OptimizationAlgorithmLevenberg(g2o::make_unique<BlockSolverType>(g2o::make_unique<LinearSolverType>()));*/std::unique_ptr<g2o::BlockSolverX::LinearSolverType> linearSolver (new g2o::LinearSolverCSparse<g2o::BlockSolverX::PoseMatrixType>());std::unique_ptr<g2o::BlockSolverX> solver_ptr (new g2o::BlockSolverX(std::move(linearSolver)));g2o::OptimizationAlgorithmLevenberg* solver = new g2o::OptimizationAlgorithmLevenberg(std::move(solver_ptr));g2o::SparseOptimizer optimizer;optimizer.setAlgorithm(solver);optimizer.setVerbose(true);/// build g2o problemconst double *observations = bal_problem.observations();// vertexvector<VertexPoseAndIntrinsics *> vertex_pose_intrinsics;vector<VertexPoint *> vertex_points;for (int i = 0; i < bal_problem.num_cameras(); ++i) {VertexPoseAndIntrinsics *v = new VertexPoseAndIntrinsics();double *camera = cameras + camera_block_size * i;v->setId(i);v->setEstimate(PoseAndIntrinsics(camera));optimizer.addVertex(v);vertex_pose_intrinsics.push_back(v);}for (int i = 0; i < bal_problem.num_points(); ++i) {VertexPoint *v = new VertexPoint();double *point = points + point_block_size * i;v->setId(i + bal_problem.num_cameras());v->setEstimate(Vector3d(point[0], point[1], point[2]));// g2o在BA中需要手动设置待Marg的顶点v->setMarginalized(true);optimizer.addVertex(v);vertex_points.push_back(v);}// edgefor (int i = 0; i < bal_problem.num_observations(); ++i) {EdgeProjection *edge = new EdgeProjection;edge->setVertex(0, vertex_pose_intrinsics[bal_problem.camera_index()[i]]);edge->setVertex(1, vertex_points[bal_problem.point_index()[i]]);edge->setMeasurement(Vector2d(observations[2 * i + 0], observations[2 * i + 1]));edge->setInformation(Matrix2d::Identity());edge->setRobustKernel(new g2o::RobustKernelHuber());optimizer.addEdge(edge);}optimizer.initializeOptimization();optimizer.optimize(40);// set to bal problemfor (int i = 0; i < bal_problem.num_cameras(); ++i) {double *camera = cameras + camera_block_size * i;auto vertex = vertex_pose_intrinsics[i];auto estimate = vertex->estimate();estimate.set_to(camera);}for (int i = 0; i < bal_problem.num_points(); ++i) {double *point = points + point_block_size * i;auto vertex = vertex_points[i];for (int k = 0; k < 3; ++k) point[k] = vertex->estimate()[k];}
}

习题

√ 题 1

证明式 (9.25) 成立。提示：你可能会用到 SMW 公式，参考文献 [6, 76].

根据 SMW(Sherman-Morrison-Woodbury) 恒等式证明 K \bm{K} K 的求解可以不依靠 P ^ k \bm{\hat{P}}_k P^k。
即用别的已知量替换 K \bm{K} K 定义中的 P ^ k \bm{\hat{P}}_k P^k

State estimation for robotics: A matrix lie group approach.
来自上述书籍的公式：

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证明:
K = P ^ k C k T Q − 1 由式 ( 9.16 ) = ( C k T Q − 1 C k + P ˇ k − 1 ) − 1 C k T Q − 1 由式 ( 2.75 c ) 由右到左其中： A ⇒ P ˇ k , B ⇒ C k T , C ⇒ C k , D ⇒ Q = P ˇ k C k T ( Q + C k P ˇ k C k T ) − 1 \begin{align*} \bm{K} & = \bm{\hat{P}}_k\bm{C}_k^T\bm{Q}^{-1} \\ & 由式 (9.16) \\ &= (\bm{C}_k^T\bm{Q}^{-1}\bm{C}_k+\bm{\check{P}}_k^{-1})^{-1}\bm{C}_k^T\bm{Q}^{-1} \\ & 由式 (2.75c) 由右到左 \\ & 其中： A \Rightarrow \bm{\check{P}}_k, B \Rightarrow \bm{C}_k^T, C \Rightarrow \bm{C}_k, D \Rightarrow \bm{Q}\\ & = \bm{\check{P}}_k\bm{C}_k^T(\bm{Q} + \bm{C}_k\bm{\check{P}}_k\bm{C}_k^T)^{-1} \end{align*} K=P^kCkTQ−1由式(9.16)=(CkTQ−1Ck+Pˇk−1)−1CkTQ−1由式(2.75c)由右到左其中：A⇒Pˇk,B⇒CkT,C⇒Ck,D⇒Q=PˇkCkT(Q+CkPˇkCkT)−1

证毕。

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